2 March 2017

Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a series of measurements observed over time, containing statistical noise and other inaccuracies, and produces estimates of unknown variables that tend to be more accurate than those based on a single measurement alone, by using Bayesian inference and estimating a joint probability distribution over the variables for each timeframe. The filter is named after Rudolf E. Kálmán, one of the primary developers of its theory.

original.estimate <- 4 original.error <- 2
current.estimate <- original.estimate current.error <- original.error
\(\text{kalman gain} = KG = \frac{E_{EST}}{E_{EST} + E_{MEA}}\)
kalman.gain = current.error / ( current.error + measured.error )
\(\text{measurement} = MEA\)
previous.estimate <- current.estimate
\[EST_t = EST_{t-1} + KG [MEA - EST_{t-1} ]\]
current.estimate = previous.estimate + kalman.gain * (measurement - previous.estimate)
\[E_{EST_t} = \frac{E_{MEA}E_{EST_{t-1}}}{E_{MEA}+E_{EST_{t-1}}} \Rightarrow [1-KG] E_{EST_{t-1}}\]
current.error = ( measured.error * previous.error ) /
( measured.error + previous.error )
current.error = (1 - kalman.gain) * previous.error